A Local Model for the Energy Transfer in a Saturated Static Mixture

In the present work the transient energy transfer in a nonsaturated porous medium is studied, using a mixture theory viewpoint. The porous matrix is assumed homogeneous, rigid and isotropic, while the fluid is a Newtonian incompressible one and both are assumed static. Since the homogeneous matrix is not saturated, gradients of concentration are present. The porous medium and the fluid (a liquid) will be regarded as continuous constituents of a mixture that will have also a third constituent, an inert gas, assumed with zero mass density and thermal conductivity. The problem is described by a set of two partial differential equations which represent the energy balances for the fluid and the solid constituents. Isovalues for these two constituents are plotted, considering representative time instants and selected values for the energy equations coefficients and for the saturation.     

A hyperbolic mathematical modeling for describing the transition saturated/unsaturated in a rigid porous medium

This work proposes a mathematical model to study the filling up of an unsaturated porous medium by a liquid identifying the transition from unsaturated to saturated flow and allowing a small super saturation. As a consequence the problem remains hyperbolic even when saturation is reached. This important feature enables obtaining numerical solution for any initial value problem and allows employing Glimm’s scheme associated with an operator splitting technique for treating drag and viscous effects. A mixture theory approach is used to build the mechanical model, considering a mixture of three overlapping continuous constituents: a solid (porous medium), a liquid (Newtonian fluid) and a very low-density gas (to account for the mixture compressibility). The constitutive assumption proposed for the pressure gives rise to a continuous function of the fluid fraction. The complete solution of the Riemann problem associated with the system of conservation laws, as well as four examples, considering all the four possible connections, namely, 1-shock/2-shock, 1-rarefaction/2-rarefaction, 1-rarefaction/2-shock and 1-shock/2-rarefaction are presented.     

Numerical simulation of flow boiling for organic fluid with high saturation temperature in vertical porous coated tube

A semi-analytical model is developed for the prediction of flow boiling heat transfer inside vertical porous coated tubes. The model assumes that the forced convection and nucleate boiling coexist together in the annular flow regime. Conservations of mass, momentum, and energy are used to solve for the liquid film thickness and temperature. The heat flux due to nucleate boiling consists of those inside and outside micro-tunnels. To close the equations, a detailed analysis of various forces acting on the bubble is presented to predict its mean departure diameter. The active nucleation site density of porous layer is determined from the pool boiling correlation by introducing suppression factor. The flow boiling heat transfer coefficients of organic fluid (cumene) with high saturation temperature in a vertical flame-spraying porous coated tube are studied numerically. It is shown that the present model can predict most of the experimental values within ±20%. The numerical results also indicate that the nucleate boiling contribution to the overall heat transfer coefficient decreases from 50% to 15% with vapor quality increasing from 0.1 to 0.5.     

Comparison of Pore-Level and Volume-Averaged Computations in Highly Conductive Spherical-Void-Phase Porous Materials

A study has been carried out to compare results obtained from pore-level simulations conducted on three-dimensional idealized spherical-void-phase geometric models to similar results obtained from a solver based on volume-averaging and local thermal non-equilibrium. The purpose of the comparison is to establish closure coefficients for the viscous and form drag terms in the volume-averaged momentum equations and the interstitial convective exchange coefficient required to couple the volume-averaged energy equations for the solid and fluid constituents. A method is also described for determining the solid-phase conduction shape factor, which is shown to be important for accurate volume-average simulation of highly conductive porous materials. The shape factor has been addressed in previous literature (using various terminology) and accounts in a bulk manner for resistance due to the elongated conduction path and for changes in the effective heat flow area along the conduction path. The conduction shape factor is a function of the geometry only and is found herein from a detailed comparison between pore-level and volume-averaged simulations of conjugate heat transfer. The conduction shape factor vastly improves volume-averaged predictions of the overall heat transfer and the temperature distributions in the porous material.     

The Effect of Double Dispersion on Natural Convection Heat and Mass Transfer in a Non-Newtonian Fluid Saturated Non-Darcy Porous Medium

The effect of power law index parameter of the non-Newtonian fluid on free convection heat and mass transfer from a vertical wall is analyzed by considering double dispersion in a non-Darcy porous medium with constant wall temperature and concentration conditions. The Ostwald–de Waele power law model is used to characterize the non-Newtonian fluid behavior. In this case a similarity solution is possible. The variation of heat and mass transfer coefficients with the governing parameters such as power law index, thermal and solutal dispersion parameters, inertia parameter, buoyancy ratio, and the Lewis number is discussed for a wide range of values of these parameters.     

Thermo-Osmosis Effect in Saturated Porous Medium

In this paper, the thermo-poroelasticity theory is used to investigate the quasi-static response of temperatures, pore pressure, stress, displacement, and fluid flux around a cylindrical borehole subjected to impact thermal and mechanical loadings in an infinite saturated poroelastic medium. It has been reported in literatures that coupled flow known as thermo-osmosis by which flux is driven by temperature gradient, can significantly change the fluid flux in clay, argillaceous and many other porous materials whose permeability coefficients are very small. This study presents a mathematical model to investigate the coupled effect of thermo-osmosis in saturated porous medium. The energy balance equations presented here fulfill local thermal non-equilibrium condition (LTNE) which is different from the local thermal equilibrium transfer theory, accounting for that temperatures of solid and fluid phases are not the same and governed by different heat transfer equations. Analytical solutions of temperatures, pore pressure, stress, displacement, and fluid flux are obtained in Laplace transform space. Numerical results for a typical clay are used to investigate the effect of thermo-osmosis. The effects of LTNE on temperatures, pore pressure, and stress are also studied in this paper.     

Influence of heat transfer and fluid flow on crack growth in multilayered porous/dense materials using XFEM: Application to Solid Oxide Fuel Cell like material design

An advanced numerical model is developed to investigate the influence of heat transfer and fluid flow on crack propagation in multi-layered porous materials. The fluid flow, governed by the Navier–Stokes and Darcy’s law, is discretized with the nonconforming Crouzeix–Raviart (CR) finite element method. A combination of Discontinuous Galerkin (DG) and Multi-Point Flux Approximation (MPFA) methods is used to solve the advection–diffusion heat transfer equation in the flow channel and in the fluid phase within the porous material. The crack is assumed to affect only the heat diffusion within the porous layer, therefore a time splitting technique is used to solve the heat transfer in the fluid and the solid phases separately. Thus, within the porous material, the crack induces a discontinuity of the temperature at the crack surfaces and a singularity of the flux at the crack tip. Conduction in the solid phase is solved using the eXtended Finite Element Method (XFEM) to better handle the discontinuities and singularities caused by the cracks. The XFEM is also used to solve the thermo-mechanical problem and to track the crack propagation. The multi-physics model is implemented then validated for the transient regime, this necessitated a post processing treatment in which, the stress intensity factors (SIF) are computed for each time step. The SIFs are then used in the crack propagation criterion and the crack orientation angle. The methodology seems to be robust accurate and the computational cost is reduced thanks to the XFEM.     

Acoustical characterization of fluid-saturated porous media with local heterogeneities: Theory and application

A linear dynamic model of fully saturated porous media with local (either microscopic or mesoscopic) heterogeneities is developed within the context of Biot’s theory of poroelasticity. Viscoporoelastic behavior associated with local fluid flow is characterized by the notion of the dynamic compatibility condition on the interface between the solid and the fluid. Complex, frequency-dependent material parameters characterizing the viscoporoelasticity are derived. The complex properties can be obtained through determining the quasi-static poroelastic parameters, the properties of individual constituents, and the relaxation time of the dynamic compatibility condition on the interface. Relationships among various quasi-static poroelastic parameters are developed. It is shown that local fluid flow mechanism is significant only in the porous media with local heterogeneities. The relaxation time of the compatibility condition on the interface depends upon the details of local structure of porous media that control local fluid pressure diffusion. The new model is used to describe the velocity dispersion and attenuation in fully saturated porous media. The proposed model provides a theoretical framework to simulate the acoustical behavior of fully saturated porous media over a wide range of frequencies without making any explicit assumption about the structure of local heterogeneities.     

A Micropolar Theory of Porous Media: Constitutive Modelling

The extension of the classical mixture theory by the concept of volume fractions leads to the theory of porous media. In this article, the theory of porous media is generalised to micropolar constituents. The kinematic relations and the balance equations for a porous medium are developed without restricting the number of constituents. Based on the entropy inequality, the general form of the constitutive equations are derived for a binary medium consisting of a porous elastic skeleton saturated by a viscous pore-fluid. Both constituents are assumed to be compressible. Handling the saturation constraint by a Lagrangian multiplier leads to a compatibility of the proposed model to so-called hybrid and incompressible models.     

A numerical study of unsteady gas-solid flow between parallel porous plates submitted to a magnetic field

This paper studies unsteady laminar flow of dusty conducting fluid between parallel porous plates with temperature dependent viscosity and the Network Simulation Method (NSM) is used to solve the governing nonlinear partial differential equations. The fluid is acted upon by a constant pressure gradient and an external uniform magnetic field is applied perpendicular to the plates that are assumed to be porous. The NSM is applied to solve the steady-state and transient problems of flow and heat transfer for both the fluid and dust particles. With this method, only discretization of the spatial co-ordinates is necessary, while time remains as a real continuous variable. The velocity and temperature are studied for different values of the viscosity and magnetic field parameters.     

Thermo-hydraulic characterization of a louvered fin and flat tube heat exchanger

In the present study, a whole heat exchanger with a hydraulic diameter of 2.3 mm is tested, which is a minichannel heat exchanger according to the Kandlikar classification. This is a louvered fin and flat tube heat exchanger currently used in car cooling systems, also known as radiator. A glycol-water mixture (60/40 in volume) circulates through the tubes at flows ranging from 100 to 7800 l/h and at a supply temperature of 90 °C. This fluid is cooled with ambient air at a temperature of 20 °C and at frontal air velocities varying between 0.5 and 7 m/s. The thermohydraulic performance of the heat exchanger is compared with the classical correlations given in the literature for the heat transfer and the friction factor calculation. On the glycol-water side the heat exchanger is characterized for Reynolds numbers from 30 to 8000. A first comparison is carried out with the correlations available in the literature with a purely predictive model by obtaining a predictive value with a systematic under prediction lower than 10%. In a second step a semi-empirical model is considered to identify the experimental heat transfer coefficients for this application.     

Thermal Non-Equilibrium Porous Convection with Heat Generation and Density Maximum

Linear stability criterion for the onset of natural convection in a fluid saturated porous medium with uniform internal heat generation and density maximum is determined. The porous medium is not in local thermal equilibrium (LTE) and we follow a two-field model for the energy equation. It is found that both the heat generation and density maximum have an additive effect in advancing the onset condition. In general the destabilising effect of density maximum increases for large values of the fluid heat generation parameter. This effect becomes prominent even for small values of the fluid heat generation parameter when the flow is of Darcy type and LTE is not valid.     

Fully-developed conjugate heat transfer in porous media with uniform heating

We propose a computational method for approximating the heat transfer coefficient of fully-developed flow in porous media. For a representative elementary volume of the porous medium we develop a transport model subject to periodic boundary conditions that describes incompressible fluid flow through a uniformly heated porous solid. The transport model uses a pair of pore-scale energy equations to describe conjugate heat transfer. With this approach, the effect of solid and fluid material properties, such as volumetric heat capacity and thermal conductivity, on the overall heat transfer coefficient can be investigated. To cope with geometrically complex domains we develop a numerical method for solving the transport equations on a Cartesian grid. The computational method provides a means for approximating the heat transfer coefficient of porous media where the heat generated in the solid varies “slowly” with respect to the space and time scales of the developing fluid. We validate the proposed method by computing the Nusselt number for fully developed laminar flow in tubes of rectangular cross section with uniform wall heat flux. Detailed results on the variation of the Nusselt number with system parameters are presented for two structured models of porous media: an inline and a staggered arrangement of square rods. For these configurations a comparison is made with literature on fully-developed flows with isothermal walls.     

Non-Darcy Mixed Convection in a Vertical Porous Channel with Boundary Conditions of Third Kind

Combined free and forced convection flow in a parallel plate vertical channel filled with porous matrix is analyzed in the fully developed region with boundary conditions of third kind. The flow is modeled using the Brinkman?CForchheimer-extended Darcy equations. The plates exchange heat with an external fluid. Both conditions of equal and different reference temperatures of the external fluid are considered. Governing equations are solved numerically by shooting technique that uses classical explicit Runge?CKutta scheme and Newton?CRaphson method as a correction scheme and analytically using perturbation series method for Darcy model. The velocity field, the temperature field and Nusselt numbers are obtained for governing parameters such as porous parameter, inertia term and perturbation parameter for equal and unequal Biot numbers and are displayed graphically. The dimensionless mean velocity and bulk temperature are also determined. It is found that the numerical solutions agree for small values of the perturbation parameter in the absence of the inertial forces.     

Thermal stretching in two-phase porous media: Physical basis for Maxwell model

An alternate yet general form of the classical effective thermal conductivity model (Maxwell model) for two-phase porous materials is presented, serving an explicit thermo-physical basis. It is demonstrated that the reduced effective thermal conductivity of the porous media due to non-conducting pore inclusions is caused by the mechanism of thermal stretching, which is a combination of reduced effective heat flow area and elongated heat transfer distance (thermal tortuosity).     

Coupled Solvent and Heat Transport of a Mixture of Swelling Porous Particles and Fluids: Single Time-Scale Problem

A three-spatial scale, single time-scale model for both moisture and heat transport is developed for an unsaturated swelling porous media from first principles within a mixture theoretic framework. On the smallest (micro) scale, the system consists of macromolecules (clay particles, polymers, etc.) and a solvating liquid (vicinal fluid), each of which are viewed as individual phases or nonoverlapping continua occupying distinct regions of space and satisfying the classical field equations. These equations are homogenized forming overlaying continua on the intermediate (meso) scale via hybrid mixture theory (HMT). On the mesoscale the homogenized swelling particles consisting of the homogenized vicinal fluid and colloid are then mixed with two bulk phase fluids: the bulk solvent and its vapor. At this scale, there exists three nonoverlapping continua occupying distinct regions of space. On the largest (macro) scale the saturated homogenized particles, bulk liquid and vapor solvent, are again homogenized forming four overlaying continua: doubly homogenized vicinal fluid, doubly homogenized macromolecules, and singly homogenized bulk liquid and vapor phases. Two constitutive theories are developed, one at the mesoscale and the other at the macroscale. Both are developed via the Coleman and Noll method of exploiting the entropy inequality coupled with linearization about equilibrium. The macroscale constitutive theory does not rely upon the mesoscale theory as is common in other upscaling methods. The energy equation on either the mesoscale or macroscale generalizes de Vries classical theory of heat and moisture transport. The momentum balance allows for flow of fluid via volume fraction gradients, pressure gradients, external force fields, and temperature gradients.     

Upscaling of the Coupling of Hydromechanical and Thermal Processes in a Quasi-static Poroelastic Medium

We undertake a formal derivation of a linear poro-thermo-elastic system within the framework of quasi-static deformation. This work is based upon the well-known derivation of the quasi-static poroelastic equations (also known as the Biot consolidation model) by homogenization of the fluid-structure interaction at the microscale. We now include energy, which is coupled to the fluid-structure model by using linear thermoelasticity, with the full system transformed to a Lagrangian coordinate system. The resulting upscaled system is similar to the linear poroelastic equations, but with an added conservation of energy equation, fully coupled to the momentum and mass conservation equations. In the end, we obtain a system of equations on the macroscale accounting for the effects of mechanical deformation, heat transfer, and fluid flow within a fully saturated porous material, wherein the coefficients can be explicitly defined in terms of the microstructure of the material. For the heat transfer we consider two different scaling regimes, one where the Péclet number is small, and another where it is unity. We also establish the symmetry and positivity for the homogenized coefficients.     

Wave scattering by flexible porous vertical membrane barrier in a two-layer fluid

The scattering of water waves by a flexible porous membrane barrier in a two-layer fluid having a free surface is analysed in two dimensions. The membrane barrier is extended over the entire water depth in a two-layer fluid, each fluid being of finite depth. In the present analysis, linear wave theory and small amplitude membrane response are assumed. The porous membrane barrier is tensioned and pinned at both the free surface and the seabed. The associated mixed boundary value problem is reduced to a linear system of equations by utilizing a general orthogonality relation along with least-squares approximation method. Because of the flow discontinuity at the interface, the eigenfunctions involved have a discontinuity at the interface and the orthogonality relation used is a generalization of the classical one corresponding to a single-layer fluid. The reflection and transmission coefficients for the surface and internal modes, the free surface and interface elevations and the nondimensional membrane deflection are computed for various physical parameters like the nondimensional tension parameter, porous-effect parameter, fluid density ratio, ratio of water depths of the two fluids to analyse the efficiency of a porous membrane as a wave barrier in the two-layer fluid.     

Upshot of ohmically dissipated Darcy-Forchheimer slip flow of magnetohydrodynamic Sutterby fluid over radiating linearly stretched surface in view of Cash and Carp method

The present work concerns the momentum and heat transmission of the electro-magnetohydrodynamic(E-MHD) boundary layer Darcy-Forchheimer flow of a Sutterby fluid over a linear stretching sheet with slip. The nonlinear equations for the proposed model are analyzed numerically. Suitable techniques are used to transform the coupled nonlinear partial differential equations(PDEs) conforming to the forced balance law, energy, and concentration equations into a nonlinear coupled system of ordinary differential equations(ODEs). Numerical solutions of the transformed nonlinear system are obtained using a shooting method, improved by the Cash and Carp coefficients. The influence of important physical variables on the velocity, the temperature, the heat flux coefficient, and the skin-friction coefficient is verified and analyzed through graphs and tables. From the comprehensive analysis of the present work, it is concluded that by intensifying the magnitude of the Hartmann number, the momentum distribution decays,whereas the thermal profile of fluid increases. Furthermore, it is also shown that by augmenting the values of the momentum slip parameter, the velocity profile diminishes. It is found that the Sutterby fluid model shows shear thickening and shear thinning behaviors.The momentum profile shows that the magnitude of velocity for the shear thickening case is dominant as compared with the shear thinning case. It is also demonstrated that the Sutterby fluid model reduces to a Newtonian model by fixing the fluid parameter to zero.In view of the limiting case, it is established that the surface drag in the case of the Sutterby model shows a trifling pattern as compared with the classical case.